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arXiv · 1308.5355

Landen transforms as families of (commuting) rational self-maps of projective space

Michael Joyce·Shu Kawaguchi·Joseph H. Silverman
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Abstract

The classical (m,k)-Landen transform F_{m,k} is a self-map of the field of rational functions C(z) obtained by forming a weighted average of a rational function over twists by m'th roots of unity. Identifying the set of rational maps of degree d with an affine open subset of P^{2d+1}, we prove that F_{m,0} induces a dominant rational self-map R_{d,m,0} of P^{2d+1} of algebraic degree m, and for 0 < k < m, the transform F_{m,k} induces a dominant rational self-map R_{d,m,k} of algebraic degree m of a certain hyperplane in P^{2d+1}. We show in all cases that R_{d,m,k} extends nicely to a map of P^{2d+1} over Spec(Z), and that {R_{d,m,0} : m \ge 0} is a commuting family of maps.

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Michael Joyce, Shu Kawaguchi, Joseph H. Silverman. 2013-08-24. Landen transforms as families of (commuting) rational self-maps of projective space. https://arxiv.org/abs/1308.5355

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